A commutator lemma for confined subgroups and applications to groups acting on rooted trees

Abstract

A subgroup H H of a group G G is confined if the G G -orbit of H H under conjugation is bounded away from the trivial subgroup in the space Sub ⁡ ( G ) \operatorname {Sub}(G) of subgroups of G G . We prove a commutator lemma for confined subgroups. For groups of homeomorphisms, this provides the exact analogue for confined subgroups (hence in particular for uniformly recurrent subgroups (URSs)) of the classical commutator lemma for normal subgroups: if G G is a group of homeomorphisms of a Hausdorff space X X and H H is a confined subgroup of G G , then H H contains the derived subgroup of the rigid stabilizer of some open subset of X X .

We apply this commutator lemma in the setting of groups acting on rooted trees. We prove a theorem describing the structure of URSs of weakly branch groups and of their non-topologically free minimal actions. Among the applications of these results, we show: (1) if G G is a finitely generated branch group, the G G -action on ∂ T \partial T has the smallest possible growth among all faithful G G -actions; (2) if G G is a finitely generated branch group, then every embedding from G G into a group of homeomorphisms of strongly bounded type (e.g. a bounded automaton group) must be spatially realized; (3) if G G is a finitely generated weakly branch group, then G G does not embed into the group IET of interval exchange transformations.